New Lower Bounds for the Number of Pseudoline Arrangements
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Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, such as the study of sorting networks. Let B_n be the number of arrangements of n pseudolines and let b_n=B_n. The problem of estimating B_n was posed by Knuth in 1992. Knuth conjectured that b_n ≤n 2 + o(n^2) and also derived the first upper and lower bounds: b_n ≤ 0.7924 (n^2 +n) and b_n ≥ n^2/6 -O(n). The upper bound underwent several improvements, b_n ≤ 0.6988 n^2 (Felsner, 1997), and b_n ≤ 0.6571 n^2 (Felsner and Valtr, 2011), for large n. Here we show that b_n ≥ cn^2 -O(n n) for some constant c>0.2083. In particular, b_n ≥ 0.2083 n^2 for large n. This improves the previous best lower bound, b_n ≥ 0.1887 n^2, due to Felsner and Valtr (2011). Our arguments are elementary and geometric in nature. Further, our constructions are likely to spur new developments and improved lower bounds for related problems, such as in topological graph drawings.
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